The geometrical structure of quantum theory as a natural generalization of information geometry Marcel Reginatto Physikalisch-Technische Bundesanstalt Braunschweig, Germany
MaxEnt 2014, Amboise, France, September 21-26, 2014
ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
1 / 20
Preliminary remarks QM has a rich geometrical structure which allows for an equivalent geometrical formulation. ◮
◮
Kibble, Geometrization of Quantum Mechanics (1979); many others. Detailed but accessible: Ashtekar and Schilling, Geometrical formulation of Quantum Mechanics (1998).
The usual approach: ◮ ◮ ◮
Start from standard QM. Identify relevant geometrical features. “Translate” the theory into a geometrical language.
Here this procedure is inverted: The geometrical structure of QM is derived from information geometry. ◮
It is a natural generalization of information geometry ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
2 / 20
Preliminary remarks What is the Geometrical formulation of QM? ◮
◮
◮
States are represented by points in a symplectic manifold (which happens to have a compatible metric). Observables are represented by certain real-valued functions on this space. The Schrödinger evolution is captured by by the symplectic flow generated by a Hamiltonian formulation.
Ashtekar and Schilling, “Geometrical formulation of Quantum Mechanics” (1998)
The work presented here relies heavily on, and extends, previous work done in collaboration with M. J. W. Hall (Centre for Quantum Dynamics, Griffith University, Brisbane, Australia). ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
3 / 20
Outline
1
Probabilities, translations, and information geometry
2
Symplectic geometry
3
Kähler geometry
4
Unitary transformations
5
Hilbert space formulation from the geometric approach
ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
4 / 20
Translations and the Fisher-Rao metric Consider an n-dimensional configuration space, x ≡ {x 1 , . . . , x n }. Probability densities P(x): P(x) ≥ 0 and
R
d n xP(x) = 1.
Translation group acting on P(x), T : P(x) → P(x + θ). The natural metric on the space of parameters is the Fisher-Rao metric (Rao, 1945), Z α 1 ∂P(x + θ) ∂P(x + θ) γjk = d nx , 2 P(x + θ) ∂θ j ∂θ k where α is a constant. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
5 / 20
The metric in the space of probabilities With a change of integration variables x → x − θ, Z α 1 ∂P(x) ∂P(x) γjk = d nx . 2 P(x) ∂x j ∂x k The line element of the Fisher-Rao metric induces a line element in the space of probability densities, Z Z α 2 n 1 ds = d x δPx δPx = d n x d n x ′ gPP (x, x ′ ) δPx δPx ′ , 2 Px where Px = P(x), δPx ≡ (∂P(x)/∂x j )∆j . We have a Riemannian geometry with metric α gPP (x, x ′ ) = δ(x − x ′ ). 2Px ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
6 / 20
Information metric: Equal distance contours on P2 Information metric (left)
Euclidean metric (right)
For discrete probabilities,
α δ gij = 2P ij i
Figures: Guy Lebanon, Riemannian geometry and statistical machine learning Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
ptb-logo
7 / 20
α δ Uniqueness of the information metric gij = 2P ij i ˇ N. N. Cencov, based on invariance under “certain probabilistically meaningful transformations” known as congruent embeddings by a Markov mapping. A simpler proof later provided by L. L. Campbell. Markov mappings can be used to map probability spaces of different dimensions; e.g.,
Figure: Guy Lebanon, Riemannian geometry and statistical machine learning Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
ptb-logo
8 / 20
Uniqueness of the information metric: The basic idea Basic idea of the proof: The inner product of any two tangent vectors must be invariant under all Markov mappings.
Figures: Guy Lebanon, Riemannian geometry and statistical machine learning Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
ptb-logo
9 / 20
Dynamics for P Consider now probabilities P(x, t) that evolve in time. Two constraints that must be satisfied at all times t : ◮ ◮
R I[P] = d n x P = 1 ⇒ I[P] a constant of the motion. P(x , t) ≥ 0.
The problem of time evolution under these constraints is solved by deriving the equations of motion from an action principle. ◮
A reasonable ansatz: A constants of the motion is often related to the invariance of a Lagrangian or Hamiltonian under a particular symmetry. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
10 / 20
Hamiltonian dynamics for P Introduce an auxiliary field S canonically conjugate to P, and the Poisson bracket Z δA δB δA δB n {A, B} = d x − . δP δS δS δP The equations of motion for P and S are δH P˙ = {P, H} = , δS
δH S˙ = {S, H} = − , δP
where H is the ensemble Hamiltonian.
Normalization of P is preserved if H does not depend explicitly on S. ◮
Implies gauge invariance under S → S + c, where c is a constant. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
11 / 20
Symplectic geometry The Poisson bracket can be written as Z δA ab δB Ω (x, x ′ ) b , {A, B} = d n x d n x ′ a δFx δFx ′ where Fxa = (Px , Sx ). The symplectic structure is 0 1 ab ′ Ω (x, x ) = δ(x − x ′ ) . −1 0 We have a symplectic structure and a corresponding symplectic geometry. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
12 / 20
A more general metric Can we extend the metric gPP (x, x ′ ), which is only defined on the subspace of probabilities P, to the space of P and S? It can be done, but certain conditions which ensure the compatibility of the metric and symplectic structures have to be satisfied. These conditions amount to requiring that the space have a Kähler structure. The natural geometry of the space of probabilities in motion is a Kähler geometry. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
13 / 20
Kähler geometry A Kähler structure brings together metric, symplectic and complex structures in a harmonious way. The Kähler conditions are Ωab = gac J cb , J ac gab J bd
= gcd ,
J ab J bc = −δac .
(1) (2) (3)
Eq. (1) : compatibility between Ωab and gab , Eq. (2) : the metric is Hermitian, Eq. (3) : Jba is a complex structure. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
14 / 20
General solution to the Kähler conditions Assume gab (x, x ′ ) = gab (x)δ(x − x ′ ), α gPS 2P x gab = δ(x − x ′ ). gSP gSS The solutions of the Kähler conditions are of the form Ωab =
gab =
J ab =
0 1 −1 0 α 2Px
Ax Ax α − 2P x
δ(x − x ′ ), Ax δ(x − x ′ ), 2Px 2) (1 + A x α 2Px 2 α (1 + Ax ) δ(x − x ′ ). −Ax
But... The functional A is not determined by the Kähler conditions! ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
15 / 20
Solution for A in the case of discrete probabilities In the discrete case, G AT gab = , A (1 + A2 )G−1 where A is an n × n matrix and GAG−1 = AT . To fix A, use the same strategy that leads to the proof of the uniqueness of the information metric. Introduce canonical transformations which generalize Markov mappings. Invariance under these “generalized Markov mappings” forces A = A 1, where A is a constant. ◮
A further canonical transformation maps the metric to the particular value A = 0. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
16 / 20
Kähler structure (A = 0): Complex coordinates With Ax = 0, the Kähler structure is given (up to a product with δ(x − x ′ )) by α 0 0 1 0 a 2P Ωab = , gab = , Jb= 2P α −1 0 0 − 2P α Define ψ =
2P α
0
.
√ √ P exp(iS/α), ψ ∗ = P exp(−iS/α).
This complex transformation leads to the standard form for a flat Kähler space, 0 iα 0 α −i 0 Ωab = , gab = , J ab = . −iα 0 α 0 0 i If α = ~, the ψ are precisely the wave functions of QM. This is a remarkable result because it is based on geometrical arguments only. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
17 / 20
Unitary transformations in the discrete case There is a symplectic structure ⇒ Sp(2n,R).
But Sp(2n,R) can not be the group of transformations of the theory. There are additional requirements: ◮ ◮
P P Normalization: i P i = i ψ i ∗ ψ i = 1. P Form invariance of the metric: dσ 2 = 2α i dψ i ∗ dψ i .
These two requirements lead to the group of rotations on the 2n-dimensional sphere, O(2n, R). But unitary transformations are the only symplectic transformations which are also rotations: Sp(2n,R) ∩ O(2n, R)= U(n). ◮
The group of unitary transformations U(n) is singled out.
Time evolution is described by a one-parameter group of unitary transformations. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
18 / 20
Hilbert space and Dirac product There is a standard construction that associates a complex Hilbert space with any infinite dimensional Kähler space (Kibble).
Z 1 ϕ d n x (φ, φ∗ ) · [g + iΩ] · ϕ∗ 2 Z 1 0 1 0 i ϕ n ∗ = d x (φ, φ ) +i 1 0 −i 0 ϕ∗ 2 Z = d n x φ∗ ϕ
hφ|ϕi =
The Hilbert space structure of quantum mechanics comes out of the Kähler geometry. The complex structure that is needed for the quantum mechanics arises in a very natural way. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
19 / 20
Summary: From information to quanta The geometrical structure of quantum theory is a natural generalization of information geometry Ingredients: ◮
The natural metric on the space of probabilities (information geometry)
◮
Time evolution via an action principle using a Hamiltonian formalism (symplectic geometry)
◮
Consistency (Kähler geometry)
◮
Invariance of the Kähler metric under “generalized Markov mappings”
None of the elements that are usually assumed to be characteristic of quantum theory are introduced a priori. ptb-logo
Marcel Reginatto (PTB)
Geometry of QM from information geometry
MaxEnt 2014
20 / 20
